Bertrand Russell once wrote that "I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere." We examine the progress of this quest for certainty in mathematics, and explore how modern developments have posed new questions about the tenuous relationship between mathematics and "truth".

Recent changes to the curriculum may require those who register for 500 level courses to typeset some of their homework assignments. Monday 5pm-6pm (August 22, 2011) in HB423 there will be an introduction to LaTeX. Outline: I. Installation Resources II. Basic Document Structure and Syntax III. Sample Homework Template IV. Brief Survey of Commutative Diagram Packages and Resources Feel free to bring your laptop in order to work examples, or ask installation questions. Those already experienced with LaTeX may be interested in developing their own homework class files like the one which will be presented, or sharing their commutative diagram tips and tricks!

3:40 pm Wednesday, August 24, 2011Virtual Topology Seminar: Projective Representations of the Mapping Class group of a surface with boundary coming from TQFT
by Charlie Frohman (University of Iowa) in HB 355

For each odd prime p, and primitive 2pth root unity, there is a projective representation of the mapping class group of a torus of dimension 2, that comes from the projective action of the mapping class group of a one punctured torus ( aka the modular group) on a portion of the state space assigned to a once punctured torus. I will prove up to conjugacy, this family extends to a continuous family of representations of the modular group defined on the unit circle. This family includes a twisted version of the canonical representation of the modular group. This means that the dilation coefficient of pseudo-anosov mapping classes can be computed as a limit of quantum invariants of mapping tori. It also means that the hyperbolic volume should also be computable, though the connection is less direct. (Joint with Joanna Kania-Bartoszynska and Mike Fitzpatrick)

Abstract: Van der Waerden's theorem (1927), a (perhaps the most) fundamental result in Ramsey theory, states that for any finite partition of the set of natural numbers there exists a cell of the partition which contains arbitrary long arithmetic progressions. We present the topological reformulation of the multidimensional van der Waerden theorem which follows from a classical result due to Furstenberg and Weiss (1978). We introduce the notion of a rational dynamical system extending the classical notion of a topological dynamical system and we prove (multiple) recurrence results for such systems via a partition theorem for the rational numbers (A.K-V. Farmaki, 2010), generalizing the previous results. Also, we give some applications of these topological recurrence results to topology, to combinatorics, to diophantine approximations and to number theory.

Starting with the work of Donaldson in the 1980s, gauge theory has been used to get nontrivial information about smooth 4-manifolds. Heegaard Floer theory is a symplectic replacement for gauge theory, with similar topological applications. Recently, the Heegaard Floer invariants (of knots, links, 3-manifolds, and 4-manifolds) were shown to be algorithmically computable. In particular, this gives a simple algorithm for detecting the genus of a knot.

After reviewing Heegaard Floer hat-homology, we'll discuss a filtration one can place on the Heegaard Floer chain complex of the 3-manifold which is the 2-fold branched covering space of the 3-sphere (branched over a knot). From this construction, one can obtain a family of knot invariants.

I describe on-going joint work with Prudence Heck in which we attempt to characterize what knots can be proved to be slice knots, in the topological category, using the current technology of 4-dim surgery. This entails trying to characterize which knots bound disks in 4-space wherein the fundamental group of the exterior is "small" (for example solvable or sub-exponential growth, or contains no free non-abelian subgroups). I will give the background and describe some of our results. This talk might be rough going for "firsties".

If a scheme X is a quotient of a smooth scheme by a finite group, it has quotient singularities---that is, it is _locally_ a quotient by a finite group. In this talk, we will see that the converse is true if X is quasi-projective and is known to be a quotient by a torus (e.g. X a simplicial toric variety). Though the proof is stack-theoretic, the construction of a smooth scheme U and finite group G so that X=U/G can be made explicit purely scheme-theoretically. To illustrate the construction, I'll produce a smooth variety U with an action of G=Z/2 so that U/G is the blow-up of P(1,1,2) at a smooth point. This example is interesting because even though U/G is toric, U cannot be taken to be toric.

A classical theorem of Kodaira states that a holomorphic line bundle on a compact complex manifold is ample if and only if it carries a metric of positive curvature. More generally, one expects the geometry of a line bundle to be reflected by the ``positivity'' of its metrics. We explore this relationship in the context of multiplier ideals.

A classical theorem of Kodaira states that a holomorphic line bundle on a compact complex manifold is ample if and only if it carries a metric of positive curvature. More generally, one expects the geometry of a line bundle to be reflected by the ``positivity'' of its metrics. We explore this relationship in the context of multiplier ideals.

We discuss spectral properties of one-dimensional Schrödinger operators of the form $H = -d^2/dx^2 + \mu$, where $\mu$ is a signed Borel measure with uniformly locally bounded total variation. Our main result is a Gordon type theorem stating that $H$ has no eigenvalues if $\mu$ can be sufficiently well approximated by periodic measures. This generalizes the work of Damanik and Stolz, where such a result was proven for $L_{1,\mathrm{loc}}$-potentials. As an application we construct Schrödinger operators with purely singular continuous spectrum.

The question of P vs. NP is one of the central questions in computer science and mathematics. (It is one of the Clay Institute Millenial Problems whose solution would yield an award of $1,000,000.) In August 2010, an HP researcher claimed to have solved the problem, using tools from mathematical logic and statistical physics, including a theorem proved by the speaker in 1982. The claim generated a huge buzz in computer science, with coverage also in the New York Times. This talk will explain what the P-vs-NP problem is, what tools were employed in the claimed proof, and what the status of the claim is.

4:00 pm Friday, September 9, 2011Geometry-Analysis Seminar: On the Existence of Stationary Solutions for some Integro-Differential Equations
by Vitali Vougalter (University of Cape Town) in HB 427

We show the existence of stationary solutions for some reaction- diffusion type equations in the appropriate H2 spaces using the fixed point technique when the elliptic problem contains second order differential operators with and without Fredholm property.

4:00 pm Monday, September 12, 2011TOPOLOGY SEMINAR: Non-triviality of knots arising from iterated infection without the use of the Tristram-Levine signature
by Chris Davis (Rice University) in HB 427

We give an explicit construction of linearly independent families of knots arbitrarily deep in the (n)-solvable filtration of the knot concordance group using first order signatures. A difference between previous constructions of infinite rank subgroups in the concordance group and ours is that the deepest infecting knots in the construction we present are allowed to have vanishing Tristram-Levine signatures."